Editing Classifying space (section)

==Formalism==
A more formal statement takes into account that ''G'' may be a [[topological group]] (not simply a ''discrete group''), and that [[Group action (mathematics)|group action]]s of ''G'' are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the [[Eilenberg–MacLane space]] construction. In homotopy theory the definition of a topological space ''BG'', the '''classifying space''' for principal ''G''-bundles, is given, together with the space ''EG'' which is the '''total space''' of the [[universal bundle]] over ''BG''. That is, what is provided is in fact a [[continuous mapping]]

:<math>\pi\colon EG\longrightarrow BG. </math>

Assume that the homotopy category of [[CW complex]]es is the underlying category, from now on. The ''classifying'' property required of ''BG'' in fact relates to π. We must be able to say that given any principal ''G''-bundle

:<math>\gamma\colon Y\longrightarrow Z\ </math>

over a space ''Z'', there is a '''classifying map''' φ from ''Z'' to ''BG'', such that <math>\gamma</math>  is the [[pullback of a bundle|pullback]] of π along φ. In less abstract terms, the construction of <math>\gamma</math> by 'twisting' should be reducible via φ to the twisting already expressed by the construction of π.

For this to be a useful concept, there evidently must be some reason to believe such spaces ''BG'' exist. The early work on classifying spaces introduced constructions (for example, the [[bar construction]]), that gave concrete descriptions of ''BG'' as a [[simplicial complex]] for an arbitrary discrete group. Such constructions make evident the connection with [[group cohomology]]. 

Specifically, let ''EG'' be the [[Delta set|weak simplicial complex]] whose ''n-'' simplices are the ordered (''n''+1)-tuples <math>[g_0,\ldots,g_n]</math> of elements of ''G''. Such an ''n-''simplex attaches to the (n−1) simplices <math>[g_0,\ldots,\hat g_i,\ldots,g_n]</math> in the same way a standard simplex attaches to its faces, where <math>\hat g_i</math> means this vertex is deleted. The complex EG is contractible. The group ''G'' acts on ''EG'' by left multiplication, 
:<math>g\cdot[g_0,\ldots,g_n ]=[gg_0,\ldots,gg_n],</math> 
and only the identity ''e'' takes any simplex to itself. Thus the action of ''G'' on ''EG'' is a covering space action and the quotient map <math>EG\to EG/G</math> is the universal cover of the orbit space <math>BG = EG/G</math>, and ''BG'' is a <math>K(G,1)</math>.<ref>{{Cite book|last=Hatcher |first=Allen |author-link=Allen Hatcher|title=Algebraic topology|date=2002|publisher=[[Cambridge University Press]]|isbn=0-521-79160-X |pages=89|oclc=45420394}}</ref>

In abstract terms (which are not those originally used around 1950 when the idea was first introduced) this is a question of whether a certain functor is [[representable functor|representable]]: the [[contravariant functor]] from the homotopy category to the [[category of sets]], defined by

:''h''(''Z'') = set of isomorphism classes of principal ''G''-bundles on ''Z.''

The abstract conditions being known for this ([[Brown's representability theorem]]) ensure that the result, as an [[existence theorem]], is affirmative and not too difficult.